Phoniton systems, devices, and methods

ABSTRACT

An artificial composite object combines a quantum of sound with a matter excitation. A phonon in a confinement structure containing the matter excites it from an initial state to an excited state corresponding to a frequency of the phonon. Relaxation of the matter back to the initial state emits a phonon of the same frequency into the confinement structure. The phonon confinement structure, for example, a cavity, traps the emitted phonon thereby allowing further excitation of the matter. The coupling between the phonon and the matter results in a quantum quasi-particle referred to as a phoniton. The phoniton can find application in a wide variety of quantum systems such as signal processing and communications devices, imaging and sensing, and information processing.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.61/613,793, filed Mar. 21, 2012, which is hereby incorporated byreference herein in its entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

This invention was made with government support under H9823006C0401awarded by the National Security Agency (NSA). The government hascertain rights in the invention.

FIELD

The present disclosure relates generally to quantum systems, and, moreparticularly, to systems, methods, and devices, for generating and usinga single cavity phonon mode coupled with a single matter excitation,also known as a phoniton.

SUMMARY

An artificial, composite particle can be composed of a cavity phononmode coupled with a matter excitation in the solid-state. In an example,the device can include a semiconductor phonon microcavity (e.g., formedfrom an epitaxially grown silicon-germanium micro-pillar heterostructureor similar) with an impurity (e.g., phosphorus atom, lithium atom, orother solid-state two-level system such as, but not limited to, aquantum dot) placed at a node or antinode of the trapped acousticphonon. The cavity phonon energy is resonant with the energy levelsplitting of the impurity. Phonitons (i.e., cavity phonon plus matterexcitation) are created either by exciting the impurity or adding aphonon (e.g., electrically, thermally, optically, or by tunneling). Theinteraction strength, phonon lifetime, and matter excitation lifetimeare such that the cavity phoniton is in the strong coupling regime. Thecavity phoniton can be employed in a variety of devices, such as, butnot limited to, single phonon sources, phonon delays and switches,phonon logic, imagers and sensors, phonon lasing, and a solid sounddevice.

In one or more embodiments of the disclosed subject matter, a quantumdevice can include a quantum two-level system within a cavity. A singlecavity phonon mode can be coupled to the quantum two-level system. Thecavity can be a phonon cavity or mechanical cavity. The two-level systemcan be formed from two appropriate energy levels of an otherwisemulti-level system (e.g., an impurity atom or a quantum dot). These twoenergy levels (e.g., the two lowest energy levels of the two-levelsystem) can be referred to as a first energy state and a second energystate of the two-level system. The cavity phonon mode can have awavelength (as well as frequency, energy, polarization and/or other modeindices) such that the energy of the phonon in the cavity is resonant(or near resonant) with a transition of the two-level system between thefirst and second energy states.

The quantum two-level system can be an impurity embedded in asemiconductor material. Alternatively, the quantum two-level system canbe an impurity, a defect, or a color center in crystalline carbon, suchas diamond, a carbon nanotube, graphene, or fullerene. In yet anotheralternative, the quantum two-level system can be a quantum dot. Thecavity can be configured to trap the phonon therein, with the quantumtwo-level system disposed at an anti-node or a node of the phonon. Thephonon can be coupled to the quantum two-level system such that thecoupling rate between the phonon and the two-level system is greaterthan losses associated with the cavity phonon mode and the two-levelsystem. The phonon can have a frequency in the terahertz regime orlower.

In one or more embodiments of the disclosed subject matter, a phonitondevice can include a crystalline host and at least one two-level system.The crystalline host can have a phonon confinement structure. The atleast one two-level system can be within the phonon confinementstructure of the crystalline host and can have at least first and secondenergy states. The phonon confinement structure can be configured todirect phonons to the at least one two-level system such that the energyof the phonons within the confinement structure corresponds to atransition frequency at or near a difference between the first andsecond energy states of the at least one impurity.

The at least one two-level system can include a donor, an acceptor, adefect (e.g., a vacancy center), a color center (e.g., a coupled vacancyplus impurity center, such as a nitrogen vacancy center in diamond),and/or a quantum dot. The phonon confinement structure can be configuredto trap the phonon therein such that an antinode or a node of the phononis at a position of the at least one impurity. The at least onetwo-level system can be a plurality of two-level systems within the samephonon confinement structure, for example, constituting a linear ortwo-dimensional array, which array can be ordered or random.Alternatively or additionally, the phonon confinement structure caninclude a plurality of cavities, with at least one impurity in eachcavity. The phonon confinement structure may be configured such thatphonons can tunnel between adjacent cavities.

The phonon confinement structure can include a cavity with a reflectorthat at least partially reflects phonons from the cavity back into thecavity. The phonon confinement structure may be configured to allowpartial escape of phonons in a coherent and directional manner. Thephoniton device may be configured as a sensor, a phonon detector, aphonon source, or any other type of quantum device, such as, but notlimited to, phonon lasers, signal processing components, terahertz (THz)imagers, switches, logic devices, and quantum information processingdevices.

For example, the crystalline host can include a semiconductor regionthat is a Group III-V or Group II-IV semiconductor. The semiconductorregion can be arranged between respective superlattices as portions ofthe phonon confinement structure. The superlattices can be formed ofGroup III-V or Group II-IV semiconductors different from those of thesemiconductor region. The at least one impurity can be a donor atom oran acceptor atom.

In another example, the crystalline host can include a semiconductorregion that is arranged between respective superlattices, which functionas portions of the phonon confinement structure. The semiconductorregion can include silicon or a Group IV semiconductor as a maincomponent. The superlattices can be formed of silicon-germanium or aGroup IV semiconductor different from that of the crystalline host. Insuch a host, the at least one two-level system can be a donor atom, suchas a phosphorus atom or a lithium atom.

In still another example, the crystalline host can be diamond, and theat least one two-level system can be a color center and/or defect. Inyet another example, the crystalline host can be a carbon nanotube, agraphene sheet, or fullerene, and the at least one two-level system canbe a donor atom, an acceptor atom, or a defect. In a further example,the crystalline host can be a levitating semiconductor nanocrystal or athree-dimensional (3D) spherical quantum dot, and the at least onetwo-level system can be a donor atom or an acceptor atom containedtherein.

In still another example, the crystalline host can be a planar membranehaving an array of holes therein as portions of the phonon confinementstructure. In such a host, the two-level system can be an acceptor atom(e.g., boron, aluminum, or indium) or a donor atom within the planarmembrane. The array of holes can be periodic or aperiodic.

In one or more embodiments of the disclosed subject matter, a method forproducing a phoniton can include trapping a phonon in a cavity, andcoupling the trapped phonon to a quantum two-level system in the cavity.The quantum two-level system can have a first energy state and a secondenergy state. The phonon can be coupled such that the phonon energy inthe cavity corresponds to a transition frequency at or near a differencebetween the first and second energy states. The quantum two-level systemcan be arranged at an anti-node or a node of the trapped phonon.

The quantum two-level system can be a solid-state two-level system. Forexample, the two-level system can include one or more of a donor, anacceptor, a color center, a defect, and a quantum dot. In anotherexample, the two-level system can include an electron gas, a hole gas,or a quantum well.

The coupling can be such that a coupling rate between the phonon and thetwo-level system is in the strong coupling regime. The trapped phononcan be coupled to the quantum two-level system such that the couplingrate between the phonon and the quantum two-level system is greater thanlosses associated with the cavity (e.g., donor relaxation and cavityloss rates). The phonon can have a frequency in the THz regime or lower.

The method can further include providing a crystalline-based devicecontaining the quantum two-level system and the resonant cavity. Priorto the trapping, the phonon can be generated external to the resonantcavity and provided thereto, or the phonon can be generated by thequantum two-level system using at least one of electrical generation,thermal generation, optical generation, or tunneling. The method canfurther include altering at least one of the first and second energystates such that the transition amplitude or transition probabilitybetween them is changed. The altering can include at least one ofstraining the material, applying an electric field, or applying amagnetic field. Alternatively or additionally, the method can furtherinclude altering a resonant frequency of the cavity such that the phononenergy in the cavity no longer corresponds to a transition frequency ator near the difference between the first and second energy states.

In one or more embodiments of the disclosed subject matter, a method forproducing a phoniton can include (a) providing a two-level system inphonon confinement structure, the impurity having first and secondenergy states, a transition between the first and second energy statescorresponding to a particular phonon mode, and (b) interacting a firstphonon with the two-level system so as to cause a transition of thetwo-level system from the first energy state to the second energy state.The method can further include (c) generating a second phonon byallowing the two-level system to transition from the second energy stateto the first energy state, and (d) re-directing the second phonon by wayof the phonon confinement structure so as to interact with the two-levelsystem to cause another transition of the two-level system from thefirst energy state to the second energy state. The method can furtherinclude repeatedly performing (c) and (d).

The two-level system can include an impurity, a donor, an acceptor, acolor center, a defect, and/or a quantum dot. Each of the first andsecond phonons can have the particular phonon mode in the phononconfinement structure. The phonon confinement structure can include areflector adjacent to a cavity containing the two-level system therein.The re-directing can include reflecting the second phonon back to thetwo-level system by way of the reflector. Additionally or alternatively,the phonon confinement structure can include a periodic or aperiodicarray of holes in a crystalline host material containing the two-levelsystem. The re-directing can include reflecting the second phonon backto the two-level system by way of the array of holes in the crystallinehost material.

Objects and advantages of embodiments of the disclosed subject matterwill become apparent from the following description when considered inconjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments will hereinafter be described in detail below with referenceto the accompanying drawings, wherein like reference numerals representlike elements. The accompanying drawings have not necessarily been drawnto scale. Where applicable, some features may not be illustrated toassist in the description of underlying features.

FIG. 1 is a schematic diagram showing generalized components of aquantum phoniton device, according to one or more embodiments of thedisclosed subject matter.

FIG. 2 is a schematic diagram of a quantum two-level system in a cavityin a quantum phoniton device, according to one or more embodiments ofthe disclosed subject matter.

FIG. 3 is an energy level diagram illustrating the splitting betweenenergy levels for cavity phoniton generation, according to one or moreembodiments of the disclosed subject matter.

FIG. 4 is a diagram illustrating the angular dependence of transverseand longitudinal coupling modes of a phonon in a cavity, according toone or more embodiments of the disclosed subject matter.

FIG. 5 is a cross-sectional view of a solid state quantum phonitondevice, according to one or more embodiments of the disclosed subjectmatter.

FIG. 6 is a schematic diagram of a device with multiple two-levelsystems within a single cavity, according to one or more embodiments ofthe disclosed subject matter.

FIG. 7A is a cross-sectional view of a solid state quantum phonitondevice with an array of donors in a single cavity, according to one ormore embodiments of the disclosed subject matter.

FIG. 7B is a cross-sectional view of a quantum phoniton device with atwo-dimensional gas in a single cavity, according to one or moreembodiments of the disclosed subject matter.

FIG. 8 is a schematic diagram showing individual phoniton cavitiescoupled together as an array, according to one or more embodiments ofthe disclosed subject matter.

FIG. 9 is a cross-sectional view of a solid state quantum phonitondevice with tunneling between cavities, according to one or moreembodiments of the disclosed subject matter.

FIG. 10 is a plan view of a two-dimensional phononic crystal structurewith acceptors placed at particular cavity sites for a quantum phonitondevice, according to one or more embodiments of the disclosed subjectmatter.

FIG. 11 is a plan view of a nanomechanical two-dimensional phononbandgap cavity for a quantum phoniton device, according to one or moreembodiments of the disclosed subject matter.

FIG. 12 is a plan view of a nanomechanical one-dimensional phononbandgap cavity for a quantum phoniton device, according to one or moreembodiments of the disclosed subject matter.

FIG. 13 shows the hole valence bands in silicon (left side, (a)) andlevel splitting in the presence of a magnetic field (right side, (b)),according to one or more embodiments of the disclosed subject matter.

FIG. 14A is a graph of the superfluid phase (SF) order parameter for amany-body phonon-qubit system involving P:Si donors, according to one ormore embodiments of the disclosed subject matter.

FIG. 14B is a graph of the SF order parameter of a many-body phononqubit system involving B:Si acceptors, according to one or moreembodiments of the disclosed subject matter.

FIG. 15 is a graph of average phonon number per site for varioustemperatures at zero hopping in a device, according to one or moreembodiments of the disclosed subject matter.

FIG. 16 is a schematic diagram of a carbon-based quantum phonitondevice, according to one or more embodiments of the disclosed subjectmatter.

FIGS. 17A-17C illustrate energy schematics of a single cavityphonon-donor, a pair of coupled cavity phonon-donors, and an infinitearray of coupled cavity phonon-donors, respectively, according toembodiments of the disclosed subject matter.

FIG. 18 is a graph showing transmission amplitude versus detuning fortwo coupled cavities, according to one or more embodiments of thedisclosed subject matter.

FIG. 19 is a graph showing the second order coherence versus hoppingfrequency for different drive strengths for a pair of coupled cavities,according to one or more embodiments of the disclosed subject matter.

FIG. 20 is a graph showing the second order coherence versus hoppingfrequency for different drive strengths for an infinite array of coupledcavities, according to one or more embodiments of the disclosed subjectmatter.

FIG. 21 is a diagram illustrating a read-out scheme from a single siteby using a homodyne/heterodyne or modified Hanbury-Brown-Twiss setup,according to one or more embodiments of the disclosed subject matter.

FIG. 22 shows level splitting in the presence of a magnetic field andstress along the 2 direction in the case of δE_(∈)>δE_(H), according toone or more embodiments of the disclosed subject matter.

DETAILED DESCRIPTION

Embodiments of the disclosed subject matter relate to various devicesand systems that generate and/or use a phoniton as well as methods forgenerating and/or using a phoniton. As used herein, phoniton refers to aquantum quasi-particle that results from the coupling between a phononwithin a confinement structure (e.g., a phonon cavity or mechanicalcavity) and a matter excitation contained in the confinement structure,in particular, the combination of a single two-level system with asingle confined phonon mode that are coupled via the electron-phononinteraction.

As used herein, two-level system (TLS) refers to an otherwisemulti-level quantum system that may be embedded in a host material(e.g., a crystalline material such as single crystal silicon) and hastwo appropriate energy levels (e.g., the lowest two energy levels) ofthe multi-level system useful for phonon generation. Examples of a TLSinclude, but are not limited to, a donor atom (e.g., phosphorus orlithium) in a silicon host material and an acceptor atom (e.g., boron,aluminum, or indium) in silicon host material. Other non-exhaustiveexamples of TLSs are provided herein. The TLS can be tuned using, forexample, an external magnetic field, an external electric field, and/ora mechanical strain. When the strong coupling regime is established, thewhole phoniton system can be tuned, for example to an external phononwaveguide.

The cavity phonon mode can have a wavelength (as well as frequency,energy, polarization and/or other mode indices) such that the energy ofthe phonon in the cavity is resonant (or near resonant) with atransition of the TLS between the lowest two energy states (or otherrelevant energy states corresponding to phonon relevant frequencies).Strong coupling between a phonon and the transition between the twoenergy levels of the TLS can be achieved by appropriate configurationand construction of a quantum device, as described in further detailherein. Moreover, devices employing acceptors as the TLS can providedispersive coupling, in particular strong dispersive coupling in the“good cavity” limit or “bad cavity” limit, as described herein.

The energy eigenstates of the phoniton are hybrid states of the TLS(i.e., two levels) and of the equidistant energy states of the confinedphonon mode (i.e., an infinite number). The combined phoniton energyspectrum is thus an infinite set of non-equidistant levels withcharacteristic Rabi splitting. Such a non-linear spectrum can beachieved by using phonons (in particular, a confined phonon mode) asmediators. As used herein, confined phonon mode refers to a standingacoustic phonon mode (or, in general, an optical phonon mode) confinedin an appropriate phonon cavity.

In a non-limiting example of a phonon cavity, a silicon micro-pillarcavity can be sandwiched between a pair of reflectors in the form ofacoustic phonon distributed Bragg reflectors. Such a configurationresults in a quasi-1D phononic band gap cavity. The micro-pillarphononic cavity can be constructed with appropriate materials,dimensions, and other characteristics such that the device operates inthe strong coupling regime. In another non-limiting example of a phononcavity, a silicon nano-membrane can have a periodic arrangement of holesin two-dimensions (2D). The cavity is formed as a plain region (i.e.,without any holes) of the membrane. In other words, the cavity is formedby a region of the membrane where the 2D periodicity of the holearrangement is altered or violated. Such a configuration results in aquasi-2D phononic band gap cavity.

As used herein, strong coupling regime refers to the state where thecoherent exchange (g) is faster than the loss rate of the TLS coherencealone and is faster than the loss rate of the confined phonon mode. Theability of a quantum coherent exchange within a time scale defined bythe inverse loss rates allows the phoniton to be used as a new elementof quantum circuit architecture. When the phoniton is designed tooperate in the strong coupling regime, it can be used, for example, as aquantum computing component with characteristics matching or exceedingthose of existing circuit QED systems.

As used herein, weak coupling regime refers to the state where thecoherent exchange is slower than the loss rate of the TLS coherence andis slower than the loss rate of the confined phonon mode. While thequantum coherent exchange is not direct in the weak coupling regime, aphoniton device operating in such a regime can be used as a singlephonon source.

As used herein, resonance (or resonant phoniton system) refers to theregime where the TLS energy splitting is equal to the energy splittingof the confined phonon mode. In a quasi-resonant regime, the TLS can beslightly frequency detuned from the cavity, or vice versa. As usedherein, the quasi-resonant regime exists if the detuning, Δ, is given by

${\Delta = {{\Omega_{TLS} - \Omega_{cavity}} < \frac{\Gamma_{TLS} + \kappa_{cavity}}{2}}},$

where Ω_(TLS) is the transition frequency of the TLS, Ω_(cavity) is thecavity frequency, Γ_(TLS) is the loss rate of the TLS, and κ_(cavity) isthe loss rate of the cavity. In other words, the condition of nearresonance (or quasi-resonance or near a transition between energy statesof the TLS) refers to the regime where the TLS-cavity mode detuning isless than the average loss rate of the phonon cavity and the TLS.

As used herein, dispersive (or dispersive phoniton system) refers to theregime where the TLS energy (or frequency) detuned from the cavitystrongly exceeds the coupling rate (g). Thus, the amount of detuningΔ>>g. In this regime, the effective TLS-cavity coupling is referred toas dispersive coupling (x) and is given by

$\chi = {\frac{g^{2}}{\Delta}.}$

Nano-membrane cavity systems employing an acceptor atom as a TLS can beused to generate such a phoniton in the dispersive coupling regime.

Referring to FIG. 1, a simplified diagram of general components of adevice for generating and/or using a phonon is shown. The device caninclude a confinement structure 106 that defines a cavity region 104containing a quantum two-level system (TLS) 102 therein. The cavityregion 104 may take the form of a crystalline host, for example, anelemental or compound semiconductor material, such as a Group IVsemiconductor (e.g., Si), a Group III-V semiconductor (e.g., GaAs), or aGroup II-VI semiconductor (e.g., CdTe). Alternatively or additionally,the crystalline host can be a crystal form of carbon, such as diamond, acarbon nanotube, a graphene sheet, or a fullerene.

The confinement structure can include various features that direct aphonon 108 of a particular frequency to the TLS 102 for interactiontherewith. The confinement structure 106 may have portions that serve todefine the cavity region 104 while other portions serve to re-directemitted or incoming phonons to the TLS 102. For example, the confinementstructure 106 can include reflectors (e.g., acoustic distributed Braggreflectors (DBR)) that reflect an emitted phonon 110 back toward the TLS102. Other features that serve to delimit the cavity region 104 can beconsidered as part of the confinement structure. A micro- or nano-pillarstructure for the cavity 104 sandwiched between top and bottom DBRreflectors can thus serve as the confinement structure. In anotherexample, the confinement structure 106 can be a nanomembrane thatincludes an array (periodic or aperiodic) of holes. In still anotherexample, the configuration of the crystalline host material (e.g., asdiamond, a carbon nanotube, graphene sheet, or fullerene) maysimultaneously define a cavity region 104 and channel phonons to a TLScontained therein. In other words, features that serve to define thecavity region 104 or to channel phonons to the TLS 102 can be part ofthe confinement structure 106.

Embodiments illustrating such examples are explained in further detailbelow. However, embodiments of the confinement structure are not limitedto the specific examples illustrated and discussed herein; rather, otherconfigurations for the confinement structure are also possible accordingto one or more contemplate embodiments, so long as the configurationsare capable of the phonon channeling and/or trapping described herein.Moreover, although certain materials have been discussed for the DBRreflectors, other materials are also possible according to one or morecontemplated embodiments. For example, when the crystalline host is aGroup III-V or Group II-VI semiconductor region, the superlattices maybe formed of other Group III-V or Group II-VI semiconductor layers. Inanother example, when the crystalline host is silicon or a Group IVsemiconductor region, the superlattices may be formed ofsilicon-germanium or other Group IV semiconductor layers that isdifferent from the crystalline host.

The TLS 102 can have at least two energy states, for example, an initialunexcited first state and an excited second state. Interaction of theTLS 102 with a phonon 108 of a particular wavelength and mode causes theTLS 102 to transition from the first state to the excited state.Relaxation of the TLS 102 back to the first state results in emission ofthe phonon 110, which can have the same wavelength and mode as phonon108, into cavity region 104. Confinement structure 106 redirects thephonon 110 back to the TLS 102, thereby allowing repeated interactionbetween the TLS 102 and the phonon. The strong coupling between thephonons and the TLS 102 results in the phoniton.

As explained in more detail below, the TLS 102 is chosen such that thetransition between the initial first state and the excited second statecorresponds to phonon frequencies of interest, in particular, phononmodes and frequencies that are trapped by the confinement structure 106.The TLS 102 can thus be strongly coupled to a particular phonon mode inthe cavity region 104, so long as the coupling rate (g) between thephonon and the TLS is much larger than the TLS relaxation (F) andassociated cavity loss rates (x). The TLS 102 can be provided in acrystalline host (e.g., a semiconductor or crystalline carbon) and be adifferent material from the crystalline host.

For example, the TLS can be an impurity, such as a donor atom or anacceptor atom, in a semiconductor material. In another example, the TLScan be a semiconductor quantum dot, or the TLS can be a donor atom or anacceptor atom in a semiconductor nanocrystal or quantum dot. In stillanother example, the TLS can be a color center or defect in thecrystalline structure of diamond. In yet another example, the TLS can bea donor atom, an acceptor atom, or defect in the crystalline structureof a carbon nanotube, graphene sheet, or fullerene. Embodimentsillustrating such examples are explained in further detail below.However, embodiments of the TLS system are not limited to the specificexamples illustrated and discussed herein; rather, other configurationsfor the TLS are also possible according to one or more contemplatedembodiments, so long as the configurations are capable of thephonon-relevant energy level splitting described herein.

FIG. 2 is a simplified illustration of a TLS 240 in a cavity 232 toassist in describing certain aspects of the disclosed subject matter. Inparticular, TLS 240 can have a first level 226 and a second level 224,which are separated by an energy amount shown as 234. Cavity 232 may bedefined by a first reflector 228 and a second reflector 230 that isdesigned to preferentially trap a single phonon mode (e.g., alongitudinal or transverse mode) at a given wavelength (λ). The TLS 240can be disposed at a node or anti-node of the phonon mode trapped withincavity 232, the interaction with the phonon 236 causing transition ofthe TLS between the first level 226 and the second level 224. For thecoupling between the phonon 236 and the TLS 240 to generate a phoniton,the transition matrix element between levels 224 and 226 should bedominated by the cavity phonon field 236 and not by other transitions,such as photon transitions. Moreover, for strong coupling between thephonon 236 and the TLS 240, the coupling rate (g) should be greater thanloss rates associated with the device, for example, TLS relaxation (F)240, cavity loss rate (x) 238, and phonon anharmonicity (anh) 242.

Silicon's unique band structure allows for a transition that is solelyphonon coupled and at an energy such that a practical device ispossible. In the case of the phosphorous donor in silicon, the lowestvalley states (A1 and T2 in FIG. 3) are the active levels; their energysplitting can be controlled by the amount of strain that the silicon isunder (due to the relaxation level of the substrate layer). This can inprinciple be tuned from 11 meV at zero strain to 3 meV at high strains.

The six-fold degeneracy due to silicon's multi-valley conduction band islifted both by applied strain (e.g., due to the lattice mismatch with asubstrate) and the sharp donor potential. For example, for a [001]compressively strained silicon region, the first excited state at zeromagnetic field of a phosphorous donor 300 (i.e., the TLS in the siliconcavity) approaches a second energy level, 304, wherein the differencefrom the lowest energy level 302 to the second energy level 304 may beΔ_(v) ^(P)=3.02 meV (0.73 THz), as shown in FIG. 3. The second energylevel 304 may be a so-called excited “valley” state. The excited valleystate 304 can have an s-like envelope function like the ground state butopposite parity. Because of this, valley state relaxation times can bemuch longer than for charge states. Energy splitting (i.e., thedifference between the first energy level 302 and second energy level304) can be controlled by the applied strain to the cavity containingthe TLS. FIG. 4 shows the angular dependence of the coupling frequency,g_(q) (O) for deformation potentials. The energy splitting implieslongitudinal (transverse) wavelengths of λ_(l)≈12.3 nm (λ_(t)≈7.4 nm).For comparison, the energy splitting to the upper 2p-like state isgreater than 30 meV, which corresponds to a wavelength of about 1.2 nmand is unlikely to be amenable to phonon cavities. Since the P:Si Bohrradius is a*_(B)≦2.5 nm in the bulk, λ>a*_(B) allows for easier TLSparticle placement (e.g., donor placement), avoiding of interfacephysics, and substantially bulk-like wave functions.

Referring to FIG. 5, an embodiment of a phoniton device having a siliconheterostructure is shown. To create the TLS, an impurity 520 is placedat the antinode (or a node) of an acoustic phonon trapped in a siliconphonon cavity 508. A strained silicon phonon cavity 508 can be grown inthe [001] direction (referenced as 506). The phonon cavity 508 can havea length or thickness, d_(c), that is approximately equal to a desiredphonon wavelength, λ. The cavity 508 can have a lateral size (i.e., in aplane perpendicular to the page of FIG. 5), D, that is enclosed by afirst acoustic distributed Bragg reflector (DBR) 510 and a second DBR514. The cavity length can be chosen to be less than a criticalthickness due to strain.

Each DBR 510, 514 can be formed as layered, epitaxially-grown, andstrain-relaxed silicon-germanium (SiGe) superlattice (SL)heterostructures, in particular, repeating pairs 512 of individuallayers. The DBR SL unit period 512 can have a first layer 516 (A layer)and a second layer 518 (B layer), where the A layers are formed ofSi_(1-x)Ge_(x) and have a thickness of d_(a), and the B layers areformed of Si_(1-y)Ge_(y) and have a thickness of d_(b), where x isdifferent than y. Although a particular number of unit layers 512 havebeen shown in FIG. 5, practical embodiments are not limited to thisnumber of layers 512 and greater or fewer layers 512 than shown in FIG.5 are also possible. The layers 516, 518 can be strain matched to asubstrate 502, which may be a Si_(1-s)Ge_(s) substrate. For example, xcan be 0.55, y can be 0.05, and s can be 0.26. An appropriate cappinglayer 504 can be provided on the second DBR 514, depending on the phononmode that is actually confined in cavity region 508.

In the case of micropillar DBR (mpDBR) structures, which are designed toincrease the phonon-TLS coupling, the DBR lateral dimension may becomecomparable to the phonon wavelength (i.e., D≧d_(c)˜λ). The confined modecan then be a mixed longitudinal/transverse one. The trapped mode withwavelength λ_(q) and phase velocity v_(g) can be designed to be resonantin energy with the first excited state of the TLS (e.g., the phosphorusdonor atom). The thickness of the SL unit cell 512 can be set to matchthe Bragg condition, i.e., d_(a,b) ^((q))=v_(a,b) ^((q))λ_(q)/4v_(q),where v_(m)) represents the phase velocities using an isotropicapproximation. The TLS system (e.g., donor atom 520) can be placed atthe center of the cavity 508 (e.g., having thickness d_(c)=λ_(q)) suchthat the displacement u(r) is maximal.

In the semi-classical picture, an acoustic phonon creates atime-dependent strain,

${{ɛ_{\alpha \; \beta}(r)} = {\frac{1}{2}\left( {\frac{\partial u_{\alpha}}{\partial r_{\beta}} + \frac{\partial u_{\beta}}{\partial r_{\alpha}}} \right)}},$

which modulates the energy bands and can drive transitions in alocalized state, e.g., a donor. For silicon, from the multi-valleyelectron-phonon interaction, the matrix element between valley states,|s,j

, can be derived as:

$\begin{matrix}{{V_{ij}^{s^{\prime}s} \equiv {\hslash \; g_{q}}} = {i{\langle{s^{\prime},{i{{{\Xi_{d}{{Tr}\left( ɛ_{\alpha\beta} \right)}} + {\frac{1}{2}\Xi_{u}\left\{ {{{\hat{k}}_{i}^{\alpha}{\hat{k}}_{i}^{\beta}} + {{\hat{k}}_{j}^{\alpha}{\hat{k}}_{j}^{\beta}}} \right\} ɛ_{\alpha\beta}}}s}},j}\rangle}}} & (1)\end{matrix}$

where {circumflex over (k)}_(i,j) are the directions toward the valleys;s, s′ label the orbital (envelope) function(s), and Ξ_(d)(Si)≈5 eV aredeformation potential constants. For the donor-phonon Hamiltonian, theinteraction (of Jaynes-Cummings type) becomes H_(g)≈h-g_(q)(σ_(ge)⁺b_(q,σ)+σ_(ge) ⁻b_(q,σ) ^(†)), where only the resonant cavity phononwith quantum numbers q, σ and energy h-ω_(q,σ) is retained, b_(q,σ) ^(†)is the phonon creation operator, and σ_(ge) ⁺≡|e

g| refers to the donor transition between ground and excited states. Inthe loss part, H_(loss)=H_(k)+H′_(anh)+H_(Γ), where H_(k) (for example,see 238 in FIG. 2) couples the cavity mode to external continuum ofother modes giving a cavity decay rate of κ=ω_(q,σ)/Q (expressed throughthe Q-factor), and H′_(anh) includes phonon decay (for example, see 242in FIG. 2) due to phonon self-interaction and also phonon scattering offimpurities (e.g., primarily due to mass fluctuations in naturalsilicon). The coupling of the TLS (e.g., donor atom) to modes other thanthe cavity mode, H_(r), leads to its spontaneous decay.

The valley states 1s(A₁), 1s(T₂) that make up the TLS are the symmetricand anti-symmetric combinations of the conduction band valley minimaalong the {circumflex over (z)}-direction (see FIG. 3). Due to oppositeparity of the states the intravalley contributions cancel. Theintervalley transitions are preferentially driven by Umklapp phononswith a wave vector q at

${q_{u} \simeq {0.3\frac{2\; \pi}{a_{o}}}},$

where q_(u)≡G₊₁−2k_({circumflex over (z)}) is the wave vector“deficiency” of the intervalleyk_({circumflex over (z)})→−k_({circumflex over (z)}) transition, and

$G_{+ 1} = {\frac{4\; \pi}{a_{o}}\left( {0,0,1} \right)}$

is the reciprocal vector along {circumflex over (z)}. Since typicalvalues give q_(u)r≈q_(u)a*_(B)=9.4>1 and qr˜1 for 3 meV, the couplingcan be calculated exactly (without using a dipole approximation) forlongitudinal and transverse polarizations as:

$\begin{matrix}{g_{q}^{(\sigma)} = {\left( \frac{a_{G}^{2}q^{2}}{2\; \rho \; \hslash \; V\; \omega_{q,\lambda}} \right)^{\frac{1}{2}}{I^{ge}(\theta)}\left\{ {\begin{matrix}{\Xi_{d} + {\Xi_{u}\cos^{2}\theta}} & \lbrack l\rbrack \\{\Xi_{u}\sin \; \theta \; \cos \; \theta} & \lbrack t\rbrack\end{matrix},} \right.}} & (2)\end{matrix}$

where

$\begin{matrix}{{a_{G} \approx 0.3},{I^{ge}(\theta)}} \\{= {\int\; {{{r\left\lbrack {\Phi_{1\; s}^{\hat{z}}(r)} \right\rbrack}^{2}}^{{- }\; {qr}}{\sin \left( {q_{u}r} \right)}}}} \\{= \frac{2\; \beta_{q}\cos \; {\theta \left( {1 - {\gamma_{q}\cos^{2}\theta}} \right)}}{{\alpha_{q}^{2}\left\lbrack {\left( {1 - {\gamma_{q}\cos^{2}\theta}} \right)^{2} - {\beta_{q}^{2}\cos^{2}\theta}} \right\rbrack}^{2}}}\end{matrix}$

is the intervalley overlapping factor,

${\alpha_{q} = {1 + {\frac{1}{4}\left( {{q^{2}a^{2}} + {q_{u}^{2}b^{2}}} \right)}}},{\beta_{q} = {\frac{1}{2\; \alpha_{q}}b^{2}{qq}_{u}}},{\gamma_{q} = {\frac{1}{4\; \alpha_{q}}\left( {a^{2} - b^{2}} \right)q^{2}}},$

and a/b are the radii of the Kohn-Luttinger envelope function Φ_(1s)^({circumflex over (z)})(r). The calculated coupling is to plane wavemodes, related to a rectangular cavity with periodic boundaryconditions. FIG. 4 shows the directionality of the coupling forlongitudinal and transverse phonons. The angular dependence in Eq. (2)for longitudinal phonons is similar to dipole emission (I_(dip)^(ge)˜cos θ), but enhanced in a cone around {circumflex over(z)}-direction due to non-dipole contributions. Uncertainties in Ξ_(d)calculations result in an overall factor of two difference in themaximal coupling.

With respect to coupling to the cavity mode, the matrix element h-g_(q)^((σ)) represents the interaction energy of the donor with the “phononvacuum field” and expresses a generic dependence ∝1/√{square root over(V)} on the normalization volume. For large volumes, the matrix elementapproaches zero. In a cavity, V is the physical volume of the mode. Byvirtue of Eq. (2), a DBR cavity can be considered with a lengthd_(c)=λ_(l)≃12.3 nm designed for longitudinal resonant phonon along thez-direction. For isotropic velocities for silicon, the following valuescan be used: v_(l)=8.99×10³ m/s, v_(t)=5.4×10³ m/s. Taking the minimallateral size D_(min)≈λ_(l) to ensure D>2a*_(B)≈5 nm results in a modevolume of V_(min)≈λ³ for P:Si. Thus, the maximal phonon-donor couplingcan be estimated as g_(1λ)=3.7×10⁹s⁻¹. For D=5λ, the coupling is stillappreciable: g_(5λ)=7.4×10⁸s⁻¹. Surface undulation typical ofstep-graded SiGe quantum wells gives D≈200 nm≦15λ, though this interfaceimperfection is avoidable with heterostructures grown on defect-freenanomembrane substrates.

For a realistic cylindrical mpDBR cavity, the modes can be constructedas standing waves with energy, h-ω, and wave number, g, along the pillarz-direction, with stress-free boundary conditions on the cylindricalsurface. The displacements for compressional modes are:u_(r)(r,z)=[A_(r)J₁(η_(l)r)+B_(r)J₁(η_(t)r)] sin qz,u_(z)(r,z)=[A_(z)J₀(η_(l)r)+B_(z)J₀(η_(t)r)] cos gz where J_(0,1)(r) areBessels of 1st kind, η_(l,t)=√{square root over (ω²/v_(l,t) ²−q²)}, andA_(i), B_(i) are constants. These modes are lower in energy and couplestrongly to the donor. The related strain has a node at the Si-cavityz-boundaries (for λ-cavity)

For a fixed resonant frequency ω and lateral size D, the dispersionrelation q=ω/v_(q)(D) has multiple solutions q_(i), i=0, 1, 2, . . . ,where q₀ stands for the fundamental mode, q₁ for the 1^(st) excitedmode, etc. Each mode propagates with its own phase velocity,v_(qi)(D)≠v_(l), v_(t). For a λ-cavity and D=λ_(l), Eq. (1) yieldsmaximal coupling to the fundamental mode with λ_(q0)=6.9 nm to be g_(1λ)⁽⁰⁾=g_(max)=6.5×10⁹ s⁻¹. As illustrated in Table 1, such a value iscomparable to the above estimation. For larger D, however, the couplingto the fundamental mode rapidly decreases (e.g., g_(5λ) ⁽⁰⁾=5.8×10⁵ s⁻¹)since the mode transforms to a surface-like Rayleigh wave(v_(Rayleigh)<v_(t)). Coupling to higher mode branches is appreciableand decreases roughly as

$\frac{1}{\sqrt{V}},$

e.g., for the 1st mode branch, g_(1λ) ⁽¹⁾=2.4×10⁹ s⁻¹ and g_(5λ)⁽¹⁾=3×10⁸ s⁻¹. Among various mode choices, for any diameter D there is ahigher excited mode with resonant wavelength close to that forlongitudinal phonons, e.g., for D=3λ_(l), the wavelength of the resonant4^(th) excited mode is λ_(q4)≃12.4 nm and the coupling g_(3λ) ⁽⁴⁾=3×10⁹s⁻¹, which values are similar to the rectangular DBR estimate.

Losses in the disclosed phoniton system are dominated by donorrelaxation and leakage of the confined phonon mode. Donor relaxation,Γ_(relax), to modes different than the cavity mode (and generally nottrapped into the cavity) is bounded by the donor spontaneous emissionrate in the bulk: Γ_(relax)≲δ. The bulk donor relaxation to longitudinaland transverse relax phonons can be calculated as Γ_(ge) ^((l))=3×10⁷s⁻¹ and Γ_(ge) ^((t))=9.2×10⁷ s⁻¹, respectively, for the 3 meVtransition (Γ=Γ_(ge) ^((l))+Γ_(ge) ^((t))=1.2×10⁸ s⁻¹). The relaxationto photons is electric dipole forbidden and suppressed by(λ_(photon)/a*_(B))²˜10¹⁰.

The cavity mode loss rate is calculated as mainly due to leakage throughthe DBR mirrors, similar to optical DBR cavities except that for phononsthere is no leakage through the sides. Generally, for the cylindricalmicropillar DBRs, the cavity mode involves coupled propagation along themicropillar of two displacement components, u_(z)(r,z), u_(r)(r,z), andtwo stress fields, T_(zz), T_(zr). For small diameters, D<<λ_(l), thefundamental mode becomes mainly longitudinal (˜u_(z)), propagating withthe Young velocity, v₀=√{square root over (E/ρ)}. Using 4×4 transfermatrices results in Q_(mpDBR) ⁽⁰⁾≃10⁶ for N=33 layers for the confined,mixed fundamental mode at D=λ_(l), which is close to the limitingQ-factor related to pure longitudinal propagation and is also similar tothe 1D DBR value relevant for D>>λ_(l). For designs proposed herein, acavity loss rate κ=Δ_(v)/h-Q≃2.8×10⁶ s⁻¹ can be obtained, which valuecan be decreased by adding more layers.

At low temperatures the phonon anharmonicity losses are negligible(e.g., a rate Γ_(anh)≃1.4×10⁴ s⁻¹ at 3 meV), while scattering offimpurity mass fluctuations in natural silicon amounts to a rate twoorders of magnitude larger, i.e., Γ_(imp)≈7×10⁵ s⁻¹. In isotopicallypurified bulk silicon (an enrichment of ²⁸Si to 99%), the scatteringrate can be reduced by an order of magnitude, and the related phononmean free path will be on the order of v_(l)/Γ_(imp)≈10 cm. In thiscase, the cavity leakage dominates (κ>>{Γ_(anh), Γ_(imp)}), and thenumber of vacuum Rabi flops can reach as high as η_(Rabi)=2 g(D)/(Γ+κ)≃102 for a cavity Q-factor of Q=10⁶, and η_(Rabi)≃77 for acavity Q-factor of Q=10⁵. For D=10λ₁ and employing a similar qualityfactor results in a η_(Rabi) ⁽¹⁾≃1(17) for the 1st (2nd) excited mode.Further, nearby modes can be well separated from the resonant mode,e.g., for the fundamental mode and D=λ_(l) the next mode (in transversedirection) is ˜0.3Δ_(v)=0.9 meV off. The transverse separation forD=10λ_(l) (for the 1st excited mode) gives 0.009Δ_(v), which is morethan two orders of magnitude larger than the linewidth Γ₀≃(Γ+κ)/2 of thetwo hybridized levels.

Table 1 shows that the donor:Si cavity-phoniton can enter the strongcoupling regime with 2 g/(Γ+κ)˜10-100. Experimental confirmation can beachieved by the observation of the “vacuum” Rabi splitting:

$\Omega_{0} = {\left\lbrack {g^{2} - \frac{\left( {\Gamma - \kappa} \right)^{2}}{4}} \right\rbrack^{\frac{1}{2}}.}$

Two resolved spectral peaks can be observed if 2Ω₀>Γ₀. The Rabisplitting can be enhanced as Ω₀≃g√{square root over (N)} by placing morethan one donor (N>1) in the cavity (e.g., via a delta-doped layer or asotherwise described herein). This could allow for large coupling evenfor large diameter micropillars/1D DBR structures (since

$\left. {{\frac{\sqrt{N}}{\sqrt{V}} \propto \frac{\sqrt{D^{2}}}{D}} = 1} \right).$

Further, strain or electric field (e.g., from a top gate) can be used totune the valley transition into or out of resonance.

Experimental techniques are available to probe the Si-phoniton (e.g., ata low temperature of T˜1K, and low phonon numbers are assumed). First,free-electron lasers can be used to probe the is −2p transitions inP:Si. Observation of the vacuum Rabi splitting is possible bymeasurement of the absorption spectrum of the allowed optically probedtransition 1s(T₂)→2p₀ (˜30 meV) using weak optical excitation.Appropriate phonons can be introduced to the system by excited valleystate emission, by piezo-actuators, or by increasing the temperature.Second, pump and probe optical techniques have been used to observecoherent phonon effects in III-V and SiGe SL heterostructures. Observingthe reflected phonons from the disclosed phoniton devices can illustratetheir phonon-Rabi splitting characteristics.

The cavity-phoniton can be realized in other materials and systems. Inparticular, the phoniton device can be compatible with high-Q phononicband-gap nano-mechanical and opto-mechanical (membrane) cavities insilicon (as discussed elsewhere herein). Quantum dots, spin transitions,color centers in diamond, and other donors (particularly Li:Si) canoffer smaller resonance energies and correspondingly larger cavities(and wavelengths). In the case of [001]-strained silicon (Si), the twolowest levels in lithium (Li) possess essentially the same statestructure as P:Si and approach a splitting of Δ_(Li)=0.586 meV for highstrain (from zero splitting at no strain). See Table 1 for thecorresponding Li:Si donor-phonon coupling for the D=λ_(l) (nowλ_(l)=63.2 nm) reference cavity. As shown in Table 1, strong couplingcan still be reached. For the DBR cavities, the Si thickness can be made80 nm to 100 nm by lowering the Ge content in the substrate. For2D-phononic bandgap cavities, direct numerical calculations forcavity-trapped phonons (ω_(r)/2π10 GHz) in Si-nanostructures show thepotential to reach Q_(cav)≧10⁷ in an ideal case.

TABLE 1 Rates and parameters for circuit QED and phoniton systems.Parameter Symbol Circuit-QED P: Si/phonon Li: Si/phonon Resonanceω_(r)/2π 10 GHz 730 GHz 142 GHz Freq. Vacuum Rabi g/ωπ 100 MHz 2.1 GHz13.8 MHz Freq. Cavity Lifetime 1/κ, Q 160 ns, 10⁴ 0.22 μs, 10⁶ 1.1 μs,10⁶ TLS Lifetime 1/Γ 2 μs 8.2 ns 22 μs Critical Atom # 2Γκ/g² <6 × 10⁻⁵<3 × 10⁻⁵ <4 × 10⁻⁵ Critical Γ²/2 g²   <10⁻⁶ <2 × 10⁻⁴ <6 × 10⁻⁷ Photon# # of Rabi 2 g/ ~100 ~102 ~93 flops (κ + Γ)

In general, a number of conditions can be satisfied in the formation ofa phoniton. First, the phonon should excite the transition between thetwo levels of the TLS directly via phonon coupling. Second, the exciteddonor state should emit into the preferred phonon mode of the cavity andnot substantially into other phonon or photon modes, which wouldotherwise be considered a loss. A phonon cavity and associatedconfinement structure should isolate the single phonon mode and trap itfor a sufficient time to allow interaction with the TLS. In addition,the phonon-matter coupling strength should be relatively large ascompared with the leakage from the cavity, disintegration of the phononinto other modes, and phonon-phonon or phonon-impurity scattering. One,some, or all of the above noted conditions may be present in embodimentsof a phoniton device.

Further details regarding the derivation of the Hamiltonian equationsand conditions for forming the phoniton can be found in U.S. ProvisionalApplication No. 61/613,793, which was filed by the instant inventors onMar. 21, 2012 and has been expressly incorporated by reference herein inits entirety. Note that in the '793 application, the term “phononiton”has been used to refer to same phonon-matter coupling as the term“phoniton” in the present application.

In an alternative quantum mode, a phoniton device can operate in thedispersive regime where the coupling rate (g) is strongly exceeded bydetuning of the TLS energy or frequency, e.g., Δ=Ω_(TLS)−Ω_(cavity)>>g.In the dispersive regime, the effective TLS-cavity coupling is given bythe dispersive coupling (χ), where

$\chi = {\frac{g^{2}}{\Delta}.}$

In some embodiments, a quantum device can be configured to operate in astrong dispersive coupling regime. For example, a nano-membrane cavitysystem incorporating an acceptor as the TLS can be configured in such amanner. In the strong dispersive regime and in the “good cavity” limit,χ>Y_(TLS)>>κ_(cavity). Thus, the phonon number states in the cavity canbe probed by using two different tones. Using one of the first tones anelectric microwave can be used to sweep around the resonance of the TLSso as to excite one of the dressed state transitions having a frequencyof Ω_(TLS)+(2n+1)χ, where n is the phonon number. The second tone can bea constant phonon tone that probes the reflectivity of the dispersivephoniton system, thereby providing a characteristic “fine structurespectrum” around Ω_(TLS).

For a dispersive readout of a TLS (such as a qubit), one can use thestrong dispersive coupling regime in the “bad cavity limit” (i.e.,γ_(TLS)κ_(cavity)) and a low phonon number regime (i.e., for n<<1) canbe used. When χ>κ_(cavity), the phonon transmission or reflectionspectrum is shifted from the cavity resonance frequency by ±χ. Thus, thequbit state is collapsed to one of the TLS eigenstates.

The concepts presented above can be extended to other quantum deviceconfigurations to create new devices or systems that generate or usemultiple phonitons. For example, multiple TLSs can be provided in asingle cavity. Such a device is shown in FIGS. 6-7B and is described inmore detail below. In another example, multiple phonon cavities can beconstructed, each with a respective TLS, such as the device shown inFIGS. 8-10 and described in more detail below. The many individualphonitons generated among the multiple phonon cavities can be coupledtogether in one, two, or three dimensions, for example, by allowingtunneling through portions of the phonon confinement structures.

Referring to FIG. 6, a generalized concept for a quantum deviceincorporating multiple TLSs in a single cavity is shown. In particular,a plurality of TLSs 620 a-620 d are disposed in a cavity definedportions 628, 630 of a phonon confinement structure, which may be, forexample, an acoustic reflector or other structure. Although only fourTLSs are shown in FIG. 6, practical embodiments of the disclosed quantumdevice can include a greater number or a fewer number of TLSs in thesame cavity. Moreover, although only a single cavity is shown, multiplecavities can be provided on the same device with one or more of thecavities having one or more TLSs contained therein.

In an example, each TLS may be disposed in the cavity at a locationcorresponding to a node or an antinode of the phonon energy trapped inthe cavity. The TLSs 620 a-620 d may thus be arrayed along a common lineor on a common plane in the cavity, although the spacing of the TLSsalong the line or within the plane need not be equal (i.e., the arraymay be periodic, aperiodic, or substantially random). Phonon interactionwith the TLSs 620 a-620 d in the cavity can result in the generation ofmultiple phonitons.

The strong coupling of the phonitons can result in interesting physicalproperties for the quantum device. For example, such a phoniton systemcan form the basis of a very low threshold phonon laser (e.g., saser)without population inversion for sound as well as other condensate-baseddevices. Typically, phonon lasing requires very high threshold due tothe very large density of states. By placing a large number of TLSs(e.g., impurities, defects, and/or quantum dots) at the antinode (ornode) of the phonon cavity and because of the strong-coupling andeffective behavior of the dilute phonitons like bosons, the quantumdevice can form a many body state, exhibiting Bose-Einsteincondensation, Mott-Insulator states, etc.

Examples of a many phoniton system are shown in FIGS. 7A-7B. In FIG. 7A,multiple TLSs 720 (e.g., an impurity such as a donor) are disposed in aone-dimensional array 730 in cavity 708. The array can be placed at thenode or antinode of the phonon energy trapped in the cavity 708 by afirst acoustic reflector 710 and a second acoustic reflector 714. Thereflectors 710, 714 may optionally be provided between a host substrate702 and a capping layer 704. In FIG. 7B, a two-dimensional electron gas760, serving as a plurality of individual TLSs, is placed at the node orantinode of the phonon energy trapped in cavity 708 by reflectors 710,714. Alternatively, the plurality of individual TLSs in FIG. 7B can beformed using a hole gas instead of the electron gas. In yet anotheralternative, the individual TLSs in the array can be a quantum well.

In the configurations of FIGS. 7A-7B, the quantum device can formmany-body states. The first acoustic reflector 710 may be configured toallow some of the phonons 706 to leak out of cavity 708. Phonons 706that escape the cavity 708 are coherent and highly directional, therebyproviding, in effect, a phonon laser, but without the need forpopulation inversion. Alternatively or additionally, these condensatescan be used for sensing applications, for example, as with atomicBose-Einstein condensates, such as gradiometers, etc.

In another embodiment, the TLS in a quantum device can take the form ofa levitating nanosphere. Operation in the strong coupling regime, evenat room temperatures, can be possible using cavities formed by suchnanospheres. However, configurations of the nanosphere are not limitedto a completely spherical geometry or the nano-scale size range. Rather,variations from a spherical geometry (e.g., cubic, egg-shaped, etc.) andnano-scale size ranges (e.g., micron scale, etc.) are also possibleaccording to one or more contemplated embodiments. Optical means, suchas external optical trapping via optical tweezers, can be used tolevitate the spheres thereby isolating the mechanical degree of freedomthereof from the environment. Other levitation mechanisms may also bepossible, such as, but not limited to, magnetic levitation. Coherentcoupling of the phoniton (i.e., phonon mode+levitating nanosphere) tothe outside network can be achieved using, for example, optomechanicalcoupling to the mechanical center-of-mass motion of the nanosphere andsubsequent coupling of the center-of-mass motion to the internalmechanical cavity mode.

In an example, mechanical cavities in the form of levitating dielectricor semiconductor nanospheres can be used. The nanospheres can be formedof silicon or diamond, for example. In still another example, thenanosphere can comprise a colloidal quantum dot. The nanospheres cancontain a TLS therein. The TLS can be, for example, a donor atom, anacceptor atom, a color center (e.g., nitrogen-vacancy center), and cancover a range of frequency (e.g., a few hundred GHz up to THz).

Referring to FIG. 8, a generalized concept for a quantum device 800incorporating multiple cavities with at least one TLS contained in eachcavity. In particular, the cavities can be arranged in respective arrays802 a-802 c (periodic or aperiodic). Although only four cavities areillustrated in each array, practical embodiments of the disclosedquantum device can include a greater number or a fewer number ofcavities. Moreover, although only three arrays are shown, practicalembodiments of the disclosed quantum device can include a greater numberor a fewer number of arrays. Although only linear arrays 802 a-802 chave been illustrate, the device 800 can be constructed as multiplelinear arrays, multiple two-dimensional arrays, a single two-dimensionalarray, multiple three-dimensional arrays, a single three-dimensionalarray, or otherwise. Moreover, although only a single TLS is shown ineach cavity, the device 800 can be constructed such that multiple TLSare in each cavity, for example, as described above with respect toFIGS. 6-7B as well as elsewhere herein.

In an example, each TLS can be disposed in its respective cavity at alocation corresponding to a node or an antinode of phonon energy trappedin the cavity. Phonon interaction with the TLS in each cavity generatesa separate phoniton. One or more of the cavities can be constructed toallow at least a portion of the phonon energy to escape, so as to allowphonons to “hop” or tunnel between cavities, for example, phonon 804travelling between adjacent cavities 806 and 808. This tunneling ofphonons between cavities can result in Mott insulator states and/orcondensed states of sound. Due to the very large coupling between theTLS and the phonon, high temperature (e.g., room temperature) operationis possible. Such a configuration of the quantum device 800 may find useas a sensor or massively parallel single phonon sources, as well as inother applications.

Referring to FIGS. 9-10, two possible quantum device designs are shownthat can be used to generate a many-body phonon state. In particular,FIG. 9 shows a quantum device 900 employing strain-matched siliconsuperlattice heterostructure 910 with acoustic DBR layers of, forexample, Si_(x)G_(1-x)/Si_(y)Ge_(1-y) with a donor atom 920 in eachsilicon cavity region 908. The superlattice heterostructure 910 canallow phonons 904 to tunnel between adjacent cavities. The device 900thus has multiple silicon cavity regions 908 disposed in a row and aseparated by alternating layers of silicon-germanium reflectors. In sucha setup, the overall reflectivity of the layers between any two adjacentsilicon cavities 908 relates to a phonon inter-cavity hopping frequency,t_(ij). A suitable donor atom 920 placed in each of these cavities 908can be strongly coupled (the regime where coupling frequency is muchlarger than the donor relaxation and cavity loss rates as describedelsewhere herein) to a specifically chosen single cavity-phonon mode ω,hence, forming a cavity-phoniton

In FIG. 10, a quantum device 1000 employs a two-dimensional phoniccrystal structure with acceptor atoms 1008 placed at cavity sites.Confinement structure 1002 can include an array of holes 1004 spacedacross the device 1000 and may be configured to allow tunneling ofphonons 1006 to reach adjacent acceptor atoms 1008. Thus, an engineereddisturbance in periodicity can be used for trapping a desired phononmode in a given region. Placement of an acceptor impurity 1008 into eachof these regions can lead to cavity-phonitons with engineeredinter-cavity tunneling.

To determine the parameter range for hopping and transition frequenciesof quantum phase transitions, an equilibrium system where the phonitonnumber density is fixed can be used, which provides a good approximationwhen the phoniton lifetime is longer than the thermalization time. Forphoniton arrays having phosphorus donors (or boron acceptors) andphonons in a silicon phononic crystal or a DBR array (for example, asshown in FIGS. 9-10), the total many-body Hamiltonian is given by thestandard Jaynes-Cummings-Hubbard (JCH) model:

H _(JCH) =H _(JC)−Σ_((i,j))t_(ij)a_(i) ^(†)a_(j),  (3)

H _(JC)=Σ_(i)[∈σ_(i) ⁺σ_(i) ⁻ +ωa _(i) ^(†) a _(i) +g(σ_(i) ⁺ a_(i)+σ_(i) ⁻ a _(i) ^(†))],  (4)

where a_(i)(a_(i) ^(†)) is the phonon annihilation (creation) operatorat a given cavity site, i, whereas σ_(i) ⁺(σ_(i) ⁻) is the excitation(de-excitation) operator of the donor at that site. The inter-cavityphonon tunneling is given by the hopping frequency t_(ij) for thenearest neighbor cavity sites i and j. The regular Jaynes-CummingsHamiltonian H_(JC) corresponds to the interaction of a single mode ofthe cavity phonon with a TLS. The fast oscillating terms (i.e., σ_(i)⁺a_(i) ^(†)) responsible for virtual transitions have been dropped viarotating wave approximation. The third term in Eq. (4) is solelyresponsible for effective, nonlinear on-site phonon repulsion.

The phase transition between a Mott insulator (MI) and a superfluidphase (SF) can be determined in the grand canonical ensemble where achemical potential pt introduced as H=H_(JCH)−μΣ_(i)N_(i) fixes thenumber density. The operator N=Σ_(i)N_(i)=Σ_(i)a_(i) ^(†)a_(i)+σ_(i)^(†)σ_(i) ⁻ defines the total number of excitations. For simplicity, therandom on-site potential with zero mean (e.g., fluctuations of thechemical potential), δμ_(i), can be assumed to vanish, and t_(ij) can beassumed to be a uniform short-range hopping. The boundary between the MIand the SF phases (Mott lobes) can be determined by the value of μ forwhich adding or removing a particle does not require any energy.Introducing the SF order parameter, ψ=

a_(i)

via mean-field theory and applying the decoupling approximation, i.e.,a_(i) ^(†)a_(j)=

a_(i) ^(†)

+a_(j)+a_(i) ^(†)

a_(j)

−

a_(i) ^(†)

a_(j)

, yields the mean-field Hamiltonian:

H _(MF) =H _(JC)−Σ_(i) {ztψ(a_(i) ^(†) +a _(i))+zt|ψ| ² −μN _(i)}.  (5)

The correlation number z is the number of nearest neighbors in a givenarray geometry.

Minimization of the ground state energy E of the mean-field Hamiltonianfor different parameter ranges of μ, ω, t for phosphorus (donor) andboron (acceptor) in silicon yields the Mott lobes illustrated in FIGS.14A-14B. In particular, FIG. 14A relates to a many-body phonon qubitsystem involving P:Si donors while FIG. 14B relates to a many-bodyphonon qubit system involving B:Si acceptors. The SF order parameter, ψ,is shown as a function of the phonon hopping frequency t and chemicalpotential μ with cavity frequency of ω. MI lobes correspond to theregions of ψ=0 where the number of phonons in each lobe is constant (

n

=0, 1, 2, . . . ). The SF phase corresponds to ψ≠0.

For the calculation of Mott lobes, in the case of a phosphorus donorimpurity, an acoustic DBR design with correlation number z=2 can beused, for example, for the configuration shown in FIG. 9. The donorvalley states 1s(A₁) and 1s(T₂) make up the TLS with a transitionfrequency of ∈=0.7 THz, which corresponds to a wavelength ofapproximately 12 nm. Due to this small wavelength, DBR heterostructurescapable of small cavity lengths can be more suitable device structuresfor maximal coupling. A large array of silicon/DBR heterostructurephonon cavities can be designed to support a fundamental longitudinalacoustic (LA) phonon mode in resonance with the donor transition (ω=∈).

In the case of the boron acceptor impurity, the transverse acoustic (TA)phonon modes of the cavities can yield maximum coupling. TA phononcavity mode of ω=14 GHz (λ=390 nm) can be in resonance with the spinsplitting (in the presence of a uniform magnetic field of B=1 T) of theboron valence band acting as a TLS. However, at this large wavelength,DBR phonon cavities may be more difficult to construct due to thecritical thickness constraints. As a result, 2D phononic crystaldesigns, such as that shown in FIG. 10, may be used to implement thequantum device.

Using a quality factor of Q=10⁵, the average phonon number

n

per site versus μ for various temperatures can be calculated. FIG. 15shows the average phonon number per site various temperatures at zerohopping. In the figure, plateaus of constant

n

correspond to MI states. The stable MI states (compressibility,

$\left. {\frac{\partial{\langle n\rangle}}{\partial\mu} = 0} \right)$

quickly shrink with increased temperatures. The maximum temperatureallowed to access the first MI state is given as T=0.04-0.06 g/k_(B) interms of coupling strength.

The MI and the SF states exhibit different coherence characteristicswhich can be accessed via coherence (correlation) function measurementsin setups similar to modified homodyne/heterodyne orHanbury-Brown-Twiss. Each of these techniques generally requires singlephonon detectors. However, even with single phonon detectorsunavailable, another useful tool, a so called phonon-to-photontranslator (PPT), can be used to coherently convert phonons to photons,therefore allowing optical detection techniques to be applied on thecavity phonon-TLS system.

The many-phoniton system can be driven at each site by a phonon field ofamplitude Ω_(i) and frequency ω_(d). Switching to the rotating frame ofthe driven field yields the time-independent Hamiltonian given by:

H _(S)=Σ_(i)[Δ∈σ_(i) ⁺σ_(i) ⁻ +Δωa _(i) ^(†) a _(i) +g(σ_(i) ⁺ a_(i)+σ_(i) ⁻ a _(i) ^(†))]−Σ_((i,j)) t _(i,j)(a _(i) ^(†) a _(j) +a _(i)a _(j) ^(†))+Σ_(i)+Ω_(i)(a _(i) ^(⇓+a) _(i))=H _(S) ⁰ +H _(S) ^(d),  (6)

where Δ∈=∈−ω_(d) (Δω=ω−ω_(d)) is the detuning between the driving fieldand the TLS (cavity). The driving field Hamiltonian is separated asH_(s) ^(d). In the case of dissipation defined by the cavity loss rate(κ) and the qubit relaxation rate (Γ), the master equation for thedensity matrix is given by:

{dot over (ρ)}=i[H_(S),ρ]κΣ_(i) L[a _(i)]ρ+δΣ_(i) L[σ _(i) ⁻]ρ,  (7)

where the Lindblad super operator is defined as L[Ô]ρ=ÔρÔ^(†)−{Ô^(†)Ô,ρ}/2. The number of elements of the density matrix ρ_(i,j) to bedetermined from Eq. (6) can be given by (2Λ+1))^(2n) ^(c) where η_(c) isthe number of cavities with a single donor/acceptor inside.

A single phoniton system (e.g., as shown in FIG. 17A) can be examinedunder different driving fields and hopping conditions, for example, bydriving and measuring the heterodyne amplitude of a single site in thecase of zero hopping (i.e., t=0) and resonance (i.e., ∈=ω). As suggestedby line 18A in FIG. 18, for weak driving field strengths smaller than acritical value Ω<Ω_(c)=(κ+Γ)/4, the system initially lies in the linearresponse regime and exhibits a Lorentzian response to the driving fieldfrequency. The critical coherent drive strength is estimated as Ω_(c)^(P)˜42 MHz and Ω_(c) ^(B)˜2 MHz for a phoniton composed of aphosphorous donor and a boron acceptor, respectively.

With increasing field strengths, this response breaks down and asuper-splitting of the phonon field amplitude can occur. Such behaviorcan be understood as a coupling of the driving field only with theantisymmetric 1^(st) dressed state ((|0, e

−|1,g

)/√{square root over (2)}) and the ground state |0, g), thereforeforming a TLS. TLS treatment will stay valid with the driving fieldstrength as long as the non-linearity of the Jaynes-Cummings Hamiltonianwill only allow single phonon excitations, preventing access to thehigher multiple excitation manifolds and thus causing a phonon blockade.

In the single cavity system (e.g., FIG. 17A), the lowest two singleexcitation energies are given by ∈₂=2ω−g√{square root over (2)} and∈₁=ω−g, respectively. This yields to the condition Ω_(i)<<g(2−√{squareroot over (2)})Ω_(i)<<∈₂−2∈₁) of single excitation only subspaces of thesystem, also known as the “dressing of the dressed states.” As thesingle phoniton system still exhibits super-splitting (Ω=(κ+Γ)/2),turning on the hopping parameter (t=0.2 g) makes the two phoniton states(one phoniton in each cavity) available to occupation. This results witha clear shift in eigen frequencies and an appearance of a third peak atthe heterodyne amplitude spectrum.

The second order coherence function can be given by

γ⁽²⁾(0)=

a^(†)a^(†)aa

/

a^(†)a

²=(

Δη

²

−

η

)/

η

²+1, where the variance is Δη=η−

n

. MI phase can be identified by

${{\gamma^{(2)}(0)} = {{1 - \frac{1}{\langle n\rangle}} < 1}},$

whereas SF phase is given by γ⁽²⁾(0)=1. For a two-coupled phoniton case(e.g., as shown in FIG. 17B), the second-order coherence function γ⁽²⁾can be calculated versus the hopping frequency for different fieldstrengths. Throughout all hopping frequencies, qubits can be keptdetuned from their encapsulating cavity mode by Δ=ω−∈=t, for example, toensure a resonance with the antisymmetric mode (lowest) of the overallcoupled cavity mode. At this detuning choice, the eigen energydifference between the ground state (GS) and the lower phoniton (LP)branch can be given by ΔE=ω−g−t. The driving field can be kept inresonance with this energy difference ω_(d)=ΔE to simulate a TLS system.However, for resonant driving purposes, this detuning is not necessary,as long as the energy difference between GS and LP can be determinedeach time hopping and/or coupling parameters are changed. Even in thecase of a strong driving field, Ω>>Ω_(c), the two phoniton systemexhibits a phonon anti-bunching. FIG. 19 shows the second-ordercoherence function γ⁽²⁾ versus the hopping frequency for drive strengthsof Ω=2Ω_(c) and Ω=5Ω_(c), both in resonance with the LP branch (i.e.,ω_(d)=ω−g−t). Donors can be detuned by the hopping bandwidth Δ=t and inresonance with the anti-symmetric cavity-phonon mode.

For large cavity arrays, the mean-field theory and density matrix masterequation can be applied together for weak coherent drive and strongcoupling regime. Starting from the Hamiltonian in Eq. (6), applicationof the mean field ψ=

a

and decoupling approximation results in:

H′ _(MF)=Σ_(i)[Δ∈σ_(i) ⁺σ_(i) ⁻ +Δωa _(i) ^(†) a _(i) +g(σ_(i) ⁺ a_(i)+σ_(i) ⁻ a _(i) ^(†))−zt(a _(i) ^(†) ψ+a _(i)ψ*−ψ²)+Ω_(i)(a _(i)^(†) +a _(i))],  (8)

in the presence of a coherent driving field. Including the cavity lossand qubit relaxation, the master equation is essentially the same as Eq.(4) but with only the driven system Hamiltonian H_(S) being replaced bythe mean-field Hamiltonian H_(MF). The SF order parameter ψ is evaluatedby the self-consistency check ψ=Tr(ρa). For phonitons composed of Pdonors, the second-order coherence function γ⁽²⁾ versus the hoppingfrequency for two different field strengths can be calculated, e.g.,Ω<<Ω_(c) and Ω=2Ω_(c)=84 MHz. For this particular donor choice, thecritical drive strength can be much smaller than the coupling strength

$\left( {{i.e.},{\left. \frac{\Omega_{c}}{g} \right.\sim 0.006}} \right)$

due to already small amounts of donor relaxation and cavity loss. For aboron B acceptor, the ratio can be estimated as

${\left. \frac{\Omega_{c}}{g} \right.\sim 0.094}.$

An infinite phoniton array (e.g., as shown in FIG. 17C) can exhibit asmooth transition from incoherent to coherent case, thus indicating aphase transition from MI to SF state by increasing the hoppingfrequency. FIG. 20 shows the second-order coherence function γ⁽²⁾ versusthe hopping frequency for an infinite array. The drive strengths ofΩ<<Ω_(c) and Ω=2Ω_(c) are shown. Donors can be detuned by the hoppingbandwidth A=2t. While FIG. 19 shows that the two coupled phonitonarrangement stays anti-bunched, employing an infinite array (or at leasta sufficiently large number) of coupled phonitons, the many-body effectscan become dominant with little effect in behavior due to changes in thedriving field.

In such coupled phoniton systems, external phononic or microwave fieldscan be used to drive the phoniton system, for example to provideread-out of the system as shown in FIG. 21. Alternatively oradditionally, the external driving field can compensate for losses,provide for read/write schemes (for example, when using the phoniton asa memory device), allow measurements of the phoniton system, and/ormanipulate rigid low temperature requirements. Practical embodimentsincorporating phoniton arrays may thus use a driving field external tothe phoniton array (e.g., generated on-device, such as on-chip, oroff-device) and conveyed to the array (e.g., by a phonon waveguide orvia tunneling) to achieve one or more of the above noted purposes.

TABLE 2 Parameters for phoniton system using a phosphorus donor or aboron acceptor in silicon. Parameter Symbol P: Si B: Si Resonance Freq.ω_(r)/2π 730 GHz 14 GHz Coupling Strength g/2π 1 GHz 21.4 MHz Wavelengthλ ~12 nm ~390 nm Cavity Lifetime 1/κ, Q 22 ns 1.14 μs TLS Lifetime 1/Γ8.2 ns 0.14 μs # of Rabi flops 2 g/(κ + Γ) ~102 ~34

The observation of QPTs in large arrays may require extremely loweffective temperatures. For P:Si, T=2-3 mK (for g=1 GHz) and for B:Si,T=40-60 μK (for g=21 MHz). Active cooling can be used in such quantumdevices. Moreover, the proposed many-body phoniton systems can be usedas quantum simulators or mediators between different quantum componentsand potentially for new quantum devices.

Referring to FIGS. 11-12, a quantum circuit element based on a singleacceptor (such as B, Al, In, etc.) embedded in a patterned siliconnano-membrane is shown. Such a nano-membrane can be compatible withoptomechanical components. The acceptor qubit shown in FIGS. 11-13 canbe tunable in a range of 1-50 GHz by an external magnetic field, anadditional electric field, or an additional strain. The regime of strongresonant and also strong dispersive coupling of the qubit to a confinedacoustic phonon is also possible.

FIG. 11 illustrates a nanomechanical 2D phoniton device 1100. Anacceptor 1108 can be disposed in a cavity region 1106 surrounded by aphonon confinement structure 1104 including an array of holes 1110surrounding the cavity region 1106. FIG. 12 illustrates a nanomechanical1D phoniton device 1200. An acceptor 1208 can be provided at a cavityregion 1206 in a beam 1214. A phonon confinement structure 1204 caninclude an array of holes 1210 surrounding an outer region, a pair ofrecesses separating the beam 1214 from the outer region, and an array ofholes 1212 in beam 1214. An on-chip phonon waveguide (not shown) canallow phonon coupling in/out of the system.

FIG. 13( a) shows the hole valence bands in silicon. There is afour-fold degeneracy at the band top (and of lowest acceptor states)corresponds to particles of spin J= 3/2 (Γ₈ representation of cubicsymmetry). Application of an external magnetic field along [0,0,1]direction results ground state splitting as shown in FIG. 13( b) Levelrearrangement can also be attained via additional strain, as shown inFIG. 22. System manipulation via electric static/microwave fields isalso possible.

Two acceptor qubit arrangements are possible based on the lifting of the4-fold ground state degeneracy via external fields. For a magnetic fieldH_(z)=(0,0,H_(z)) along the crystal [0,0,1] growth direction one canchoose the lowest two Zeeman levels, |φ₁

=| 3/2

, |φ₂

=|½

, as the qubit. The Zeeman type interaction is given by:H_(H)=μ_(B){g′₁JH+g′₂(J_(χ) ³H_(χ)+c.p.)}, where c.p. is the cyclicpermutation of x, y, z; J_(X), . . . , etc. are the spin 3/2 matrices(in the crystal directions), and the renormalized g-values g′₁, g′₂(μ_(B) is Bohr magneton), depending on the acceptor bound states,fulfill the relations |g′₁|≈1, |g′₂|<<|g′₁|. The qubit energy splittingδE_(h)≈μ₀ g′₁H is tunable in the range≈1-40 GHz for H=0.1-3T. The qubicterm in H_(H) lifts the level equidistancy. The lowest and highestsplitting (see FIG. 13) are larger than the middle splitting by

$\frac{3\; g_{2}^{\prime}}{g_{1}^{\prime}} \simeq {0.09.}$

For a magnetic field H tilted away from the crystal axis [0,0,1], thequbit splitting is weakly angle dependent.

Alternatively, a second qubit arrangement is possible with mechanicalstress in addition to the magnetic field. Mechanical stress lifts theground state degeneracy only partially, e.g., for stress along thecrystal growth {circumflex over (z)}-direction, states |±3/2

and |±½

remain degenerate. Providing the stress causes an energy splittinglarger than the splitting due to magnetic field H_(z), the levelconfiguration in FIG. 13( b) rearranges so that the lowest (qubit)levels will be |φ′₁

=|−½

, |φ′₂

=|½

, as shown in FIG. 22. This forms an alternate qubit, decoupled fromphonons to first order; however, coupling can be switched on viaelectric field, as described further below. The effect of strain isdescribed by the Bir-Pikus Hamiltonian:

$\begin{matrix}{H_{ɛ} = {{a^{\prime}{Tr}\; ɛ_{\alpha \; \beta}} + {b^{\prime}ɛ_{xx}J_{x}^{2}} + {\frac{d^{\prime}}{\sqrt{3}}ɛ_{xy}\left\{ {J_{x}J_{y}} \right\}_{+}} + {c.p.}}} & (9)\end{matrix}$

The renormalized deformation potential constants a′, b′, d′ for acceptorB:Si can be estimated in the effective mass approximation (EMA), orextracted from experiment. The latter gives b′=−1.42±0.05 eV,d′=−3.7±0.4 eV. Using H_(∈), a splitting of δE_(∈)≈1-10 GHz results forexternal stress in the range of 10⁵-10⁶ Pa. Such stress can be createdin tensioned nanomechanical structures/membranes, improving also themechanical Q-factor. A much larger stress (≧10⁷ Pa) is possible due to anearby (random) crystal defect or for [0,0,1] strain due to a SiGesubstrate. The resulting splitting in the few hundred GHz rangeeffectively suppresses the qubit-phonon coupling.

In order to calculate the qubit-phonon coupling, a quantized phononfield can be considered in addition to any classical field. The couplingof the qubit {1=|3/2

, 2=|½

}, is calculated to a plane wave mode ∈_(vac)ξ_(q) ^((σ))e^(−iq·r) withpolarization ξ_(q) ^((σ)) and energy h-v_(σ)q that proved to be a goodestimation of coupling to modes with realistic boundary conditions. Therelevant matrix element is proportional to the “phonon vacuum field”strain

${ɛ_{vac} = \left( \frac{\hslash}{2\; \rho \; {Vv}_{0}q} \right)^{1\text{/}2}},$

where ρ is the mass density, v_(σ) is the sound velocity, and V is themode volume. For longwave acoustic phonons, (when the acceptor is placedat maximum strain of the nanomechanical cavity) the coupling can begiven by:

$\begin{matrix}{{g_{\sigma}^{{3\text{/}2},{1\text{/}2}} = {{d^{\prime}\left( \frac{\hslash \; \omega_{s^{\prime}s}}{8\; \rho \; \hslash^{2}{Vv}_{\sigma}^{2}} \right)}^{1\text{/}2}\begin{Bmatrix}{{\cos \; \theta},{\sigma = t_{1}}} \\{{\; \cos \; 2\; \theta},{\sigma = t_{2}}} \\{{{- }\; \sin \; 2\; \theta},{\sigma = l}}\end{Bmatrix}^{{- }\; \phi}}},} & (10)\end{matrix}$

where the polar angles of the wave vector q are with respect to{circumflex over (z)}-direction. Thus, the mode t₂ has a maximum alongthe phonon cavity (θ≈π/2). An alternative is to have an in-planemagnetic field H_(X) along the crystal [1,0,0] {circumflex over(χ)}-direction (the latter is chosen to be along the phonon cavity).Both modes t₁, t₂ (now at θ≈0) can be coupled to the cavity. The maximalcoupling g_(max,σ) ^(3/2,1/2) scales as √{square root over(ω_(S′S)/V)}∝√{square root over (q/V)}, as expected for a (1s→1s)transition. Taking a cavity V≃dλ²(d=200 nm being an exemplary thicknessfor the Si membrane) results in a coupling in the range g/2π≃0.4-21 MHzfor the frequencies of 1-14 GHz. See Table 3 below. The other allowedtransition |3/2

→|−½

(at twice the qubit frequency) is well detuned, while the transitions|3/2

→|−3/2

,|½

|−½

are phonon forbidden.

When the in-plane magnetic field has some angle θ₀ with the cavity(crystal x-axis), all transitions are allowed. Also, the qubit couplingto a confined cavity phonon will change. As a qualitative example, for aplane wave transverse mode t₁ (or t₂) along the x-axis (θ≈0), thecoupling will change in the same way as in Eq. (10), with θ replaced byθ₀. This allows manipulation of the qubit-cavity coupling by rotation ofthe magnetic field.

For the qubit {1=|3/2

, 2=|½

}, the relaxation in the cavity is bounded by the bulk phonon emissionrate, i.e., δ_(relax)≦Γ, and is given by:

$\begin{matrix}{{\Gamma_{12}\left( \theta_{0} \right)} = {\frac{E_{12}^{3}}{20\; \pi \; \rho \; \hslash^{4}}\left\{ {{{d^{\prime \; 2}\left( {{\cos^{2}2\; \theta_{0}} + 1} \right)}\left\lbrack {{{2/3}\; v_{l}^{5}} + {1/v_{t}^{5}}} \right\rbrack} + {b^{\prime \; 2}\sin^{2}2{\theta_{0}\left\lbrack {{2/v_{l}^{5}} + {3/v_{t}^{5}}} \right\rbrack}}} \right\}}} & (11)\end{matrix}$

Here the contribution of longitudinal phonons is only a few percent andresults in Table 3 are for θ=0. Note that the coupling in this case canbe switched off (e.g., for a t₁-mode along {circumflex over(χ)}-direction, at θ=π/2) while the relaxation cannot.

For the alternate qubit, {|−½

,|½

}, the stress and magnetic field are parallel along the growth{circumflex over (z)}-direction. In this scenario, both coupling andrelaxation are zero in the absence of an electric field and can beswitched on using non-zero electric field E_(z) in the same direction.The qubit-phonon coupling can be given by Eq. (10) multiplied by acoupling factor, which is a function of the splitting ratios given by

${r_{h} \equiv \frac{\delta \; E_{H}}{\delta \; E_{ɛ}}},{{{r_{e}h} \equiv {\frac{\delta \; E_{H}}{\delta \; E_{ɛ}}\text{:}\mspace{14mu} {f\left( {r_{h},r_{e}} \right)}}} =}$

(√{square root over (z₊z⁻)}−1)/√{square root over ((1+z₊)(1+z⁻))}{squareroot over ((1+z₊)(1+z⁻))}, with z_(±)=(1±√{square root over ((1±r_(h))²+r_(e) ²)}±r_(h))²/r_(e) ². Thus, for example, for r_(h)=0.5-0.9, thisfactor reaches ≈0.25-0.65 for some optimal value of the electric fieldsplitting, r_(e)≲1, which allows strong qubit-cavity coupling.

The relaxation times shown in Table 3 are comparable to T₁, T₂measurements in bulk Si at low B:Si doping (e.g., 8×10¹² cm³ or 500 nmacceptor spacing), where T_(1,exp)≃7.4 μs and T*_(2,exp)≃2.6 μs. Whilebulk T*₂ is limited by electric-dipole inter-acceptor coupling, both T₁,T₂ may be improved by the use of substantially defect-free samples,isotopically purified Si, and single acceptors. T₁ may also be improvedin nanomembranes (d<<λ) due to phase-space suppression. Finally, therelaxation to photons is strongly suppressed since (v_(t)/c)³≈10⁻¹⁵.

In the 1D/2D-phononic bandgap Si nanomembranes illustrated in FIGS.11-12, the main cavity loss mechanism is due to (fabrication)symmetry-breaking effects, coupling the cavity mode to unconfined modes,and also due to cavity surface defects. Bulk loss mechanisms may beconsidered negligible in the GHz frequency range. The cavity Q-factorfor frequencies of 1 GHz up to tens of GHz can be engineered in therange of 10⁴-10⁵, or higher.

Table 3 below shows calculations for various quantum devices. As Table 3illustrates, resonant coupling is possible for g₁₂>>Γ₁₂, κ in a widefrequency range, thereby allowing a number of Rabi flops, e.g., ˜30-100.A frequency of 1 GHz is the limit for T≃20 mK, but active cavity coolingcan be used to alter the limit. Different energy scaling of the couplingand qubit relaxation can set a maximal energy splitting of

$E_{12}^{\max} = {{\frac{5}{4\; d^{\prime}}\sqrt{\frac{\rho \; \hslash^{3}v_{t}^{5}}{\pi}}} \approx {200\mspace{14mu} {{GHz}.}}}$

At such high frequencies the limiting factor would be the significantdeterioration of the cavity Q-factor. Yet, for example, at 14 GHz even aQ=10³ can still lead to strong coupling.

Embodiments of the disclosed phoniton devices and systems can allow foron-chip manipulation of coherent acoustic phonons via coupling toacceptor qubit states in a nanomechanical cavity. Hybridization of thephonon-acceptor system and strong dispersive coupling are both possiblewith parameters comparable to circuit QED and far surpassingsemiconductor quantum dot QED. The phoniton component can beincorporated in more complex systems, for example, with phonon-photoninterfaces to photonics and in arrays of other phoniton systems forengineered many-body phonon devices. From the perspective of qubits, theisolated acceptor provides a potentially robust two-level system forquantum information processing. The phoniton components described hereinoffer an avenue for phonon dispersive readout of acceptor qubit statesand the potential for spin qubit-to-photon conversion in silicon.

The role of an atom can be taken by a single impurity in a host crystal,e.g., in silicon. The impurity-acoustic phonon interaction H_(e,ph)^(ac)(r)=Σ_(ij){circumflex over (∈)}_(ij)(r) may lead to the strongcoupling regime even for cavity effective volume of few tens of ˜λ³,since the deformation potential matrix elements are large:

Ψ_(s′)|{circumflex over (D)}_(ij)|Ψ_(s)

˜eV. Qualitatively, the large coupling can be traced from the muchsmaller bandgap (˜eV) in the “Si-vacuum” as compared to QED (˜10⁶ eV).

With respect to silicon, the 4-fold degeneracy (as shown in FIG. 13) atthe top of the valence band (neglecting heavy-light hole splitting)corresponds to propagation of particles of spin J=3/2, reflecting the Γ₈representation of cubic symmetry. Relatively large spin-orbit couplingimplies a 2-fold degenerate band (Γ₇ representation), split-off by anenergy gap Δ_(SO)≃45 meV. For shallow acceptor centers in silicon (e.g.,B, Al, In, etc.) the ionization energy, E_(A)˜Δ_(SO), so that allvalence bands can play a role in the acceptor states. Still, the lowestacceptor states remain 4-fold degenerate since the acceptor spherical(Coulomb) potential does not change the cubic symmetry of the hostcrystal.

The degeneracy of the ground state can be lifted by external magneticfield via the Zeeman type interaction H_(H) (as shown in FIG. 13),and/or via mechanical strain (as shown in FIG. 22). The relatedHamiltonians are invariants of the cubic symmetry group O_(h)=T_(d)×land time reversal and are constructed from the momentum operator,

$k_{\alpha} = {{\frac{1}{i}\frac{\partial\;}{\partial x_{a}}} + {\frac{e}{c}A_{\alpha}}}$

(or the corresponding fields) and the spin-3/2 operators J_(α), α=x, y,z.

For a relatively weak electric field, E, the linear Stark effect ispossible:

$\begin{matrix}{{H_{E} = {\frac{p_{E}}{\sqrt{3}}\left( {{E_{x}\left\{ {J_{y}J_{z}} \right\}_{+}} + {E_{y}\left\{ {J_{z}J_{x}} \right\}_{+}} + {E_{z}\left\{ {J_{x}J_{y}} \right\}_{+}}} \right)}},} & (12)\end{matrix}$

since an ion impurity actually reduces the cubic symmetry (T_(d)×l) toT_(d), and thus, there is no invariance under inversion. The groundstate splits to two doubly degenerate levels; however, the H_(E) doesnot commute with J_(z) for any direction of the field E, leading tomixing of the Zeeman states. The latter can be useful to switch on/offthe phonon coupling of the alternate qubit, {|½

,|½

}, (see FIG. 22), provided the splitting δE_(E)=2p_(E)|E| is of theorder of that due to stress, e.g., in the GHz range. The transitionelectric dipole moment, ρ_(E), can be extracted from experiments. Forexample, bulk dielectric absorption measurements can yield values ofρ_(E)≃0.26D where D=3.336×10⁻³⁰ C m is the Debye unit for e.d.m. Thus, asplitting of 1 GHz can be obtained using an electric filed|E|_(1GHz)≃3.85×10⁵ V/m, which is readily achievable in nanodevices.Note, however, that increasing the field (splitting) exponentiallydecreases the qubit lifetime due to acceptor ionization. Thus, for δE=1GHz the lifetime is τ_(ion)≈12 s, while for δE≃1.26 GHz, the lifetimereduces to τ_(ion)≈12 ms.

These numbers show that there is an experimental “window” for thealternate qubit, {|½

,|½

}, discussed above. For example, for a qubit (Zeeman) splitting ofδE_(H)=1 GHz and a strain splitting δE_(∈)=1.43 GHz (ratio of

$\left. {{r_{h} \equiv \frac{\delta \; E_{H}}{\delta \; E_{ɛ}}} = 0.7} \right),$

the coupling factor reaches a maximal value of f(r_(h),r_(e))≃0.4 forδE_(E)=1 GHz, i.e., r_(h)=0.7. Analogously, for a qubit splitting ofδE_(H)=2 GHz, the electric field splitting leads to the same couplingfactor of 0.4, thereby allowing for the possibility of a strongacceptor-phonon coupling and a relatively long (static field) ionizationlifetime.

For low-energy acoustic phonons the interaction Hamiltonian, Ĥ_(ph), hasthe form of Eq. (9) discussed above, with the strain operator

${ɛ_{ij}(r)} = {\frac{1}{2}\left( {\frac{\partial u_{i}}{\partial r_{j}} + \frac{\partial u_{j}}{\partial r_{i}}} \right)}$

expressed via the quantized mechanical displacement:u(r)=Σ_(q,σ)(u_(qσ)(r)b_(qσ)+u*_(qσ)(r)b_(qσ) ^(†)). The modenormalization is given by

${{\int{{^{3}r}\; {y_{q\; \sigma}^{*}(r)}{u_{q\; \sigma}(r)}}} = \frac{\hslash}{2\; \rho \; \omega_{q\; \sigma}}},$

so that b_(qσ) ^(†) creates a phonon in the mode (q, σ) with energyh-ω_(qσ) in a mode volume V where ρ represents the material massdensity. The vector q denotes a collective index of the discrete phononmode defined via the phonon cavity boundary conditions and mode volumeV. The phonon-acceptor coupling h-g_(qσ) ^(s≡)

^(s′;qσ|H) _(ph)|s

enters in a Jaynes-Cummings Hamiltonian such that H_(g)≈h-g_(qσ)^(s′s)(σ_(s′s) ⁺b_(q,σ)+σ_(s′s) ⁻b_(q,σ) ^(†)), where only the resonantcavity phonon has been retained and σ_(s′s) ⁺≡|s′

s| refers to the relevant acceptor transition.

TABLE 3 Rates and parameters for circuit QED, quantum dot, and phonitonsystems in a patterned silicon membrane device. Circuit- Quantum B: SiB: Si B: Si B: Si Parameter Symbol QED Dot QED (1 GHz) (4 GHz) (8 GHz)(1 Tesla) Resonance ω_(r)/2π 5.7 GHz 325 THz 1 GHz 4 GHz 8 GHz 14 GHzFreq. Vacuum Rabi g/ωπ 105 MHz 13.4 GHz 0.41 MHz 3.27 MHz 9.26 MHz 21.4MHz Freq. Cavity Lifetime 1/κ, Q 0.64 μs, 10⁴ 5.5 ps, 15.9 μs, 10⁵ 4 μs2 μs 1.14 μs 1.2 × 10⁴ Qubit Lifetime 1/Γ 84 ns 27 ps 386.5 μs 6 μs 0.75μs 0.14 μs Critical Atom # 2Γκ/g² <8.6 × 10⁻⁵ <1.87 <4.9 × 10⁻⁵   <2 ×10⁻⁴ <3.9 × 10⁻⁴ <6.9 × 10⁻⁴ Critical Photon # Γ²/2 g² <1.6 × 10⁻⁴ <9.4× 10⁻² <5.1 × 10⁻⁷ <3.2 × 10⁻⁵ <2.6 × 10⁻⁴ <1.4 × 10⁻³ # of Rabi flops 2g/(κ + Γ) ~98 ~0.8   ~79 ~99 ~64 ~34 Cavity Volume V 10⁻⁶ λ³ — 0.037 λ³0.148 λ³ 0.296 λ³ 0.52 λ³ Wavelength λ 5.26 cm 921 nm 5400 nm 1350 nm675 nm 385 nm Dispersive χ ≡ g²/Δ 17 MHz — 0.04 MHz 0.33 MHz 0.93 MHz2.14 MHz Coupling Peak resolution 2χ/Γ  ~6 — ~199 ~25  ~9  ~4 # of peaks2χ/κ ~70 —  ~8 ~16 ~23 ~31

A phonon-photon translator (PPT) can be based on optomechanicalnon-linearities that couple in the same bandgap cavity two photon modes(â, â_(p)) and a phonon mode {circumflex over (b)} via optomechanicalcoupling h_(om). For photons in the near-infrared range (i.e.,λ_(opt)≈1500 nm), the PPT can allow a quantum optical input/outputchannel (e.g., of frequency ω/2π≃200 THz) to be coupled to a phononchannel (e.g., with frequency ω_(d)/2π≃4-8 GHz). The coupling betweenthe fields can be enhanced by the auxiliary photon pump channel, pumpingat the sideband resolved frequency ω_(p)=ω−ω_(d)−Δ. At pump detuningΔ=0, it is as resonance with the red side-band of mode ω. The coherentnature of the PPT can be described by the effective beam-splitter typeHamiltonian:

H _(b-s) =−Δ{circumflex over (b)}^(†){circumflex over (b)}+G _(om)({circumflex over (a)}^(†){circumflex over (b)}+â{circumflex over(b)}^(†)),  (13)

where G_(om)∝h_(om)E₀ is the enhanced effective coupling, proportionalto the pump field amplitude, E₀. In the weak coupling regime, totaloptical reflection can be avoided when G_(om)<κ^(opt), and optimaltranslation (e.g., close to 100%) can take place at a matching conditionof G_(om)=√{square root over (κ^(opt)κ^(mech))}, where κ^(opt) andκ^(mech) are the couplings of the PPT to respective photon/phononwaveguides.

Since the PPT can be realized on the same device, e.g., on the samesilicon nanomembrane, a simultaneous photonic/phononic bandgap structureis implied. As a result, it may be considered that photons (e.g., 200THz photons) may affect the qubit lifetime when they reach the acceptor.However, the corresponding photon energy (e.g., 0.82 eV) is less thanthe indirect bandgap in silicon (e.g., ΔE_(gap)=1.1 eV). Thus, interbandtransitions are not generally possible. Considering an “ionizationprocess” of a bound hole going to the continuous spectrum (e.g., ananalog of the ionization of an (anti)hydrogen atom), the correspondingcross section is re-scaled using the following re-scaled values: freehole mass m_(A)≃0.23 m_(e) in silicon, an effective Bohr radius

${a_{A}^{eff} = \frac{^{2}Z}{{2\left\lbrack {4\; \pi \; ɛ_{0}ɛ_{r}} \right\rbrack}E_{A}}},$

with the acceptor ionization energy for B:Si of E_(A)≈0.044 eV, and ascreening factor of Z≃1.4. The total cross section can be given by:

$\begin{matrix}{{\sigma_{phot} = {\frac{32\; \pi}{3}\frac{\hslash^{6}}{c\sqrt{2\; m_{A}}{m^{3}\left\lbrack a_{A}^{eff} \right\rbrack}^{5}{E_{f}^{2.5}\left( {E_{f} + E_{A}} \right)}}\frac{1}{\left\lbrack {4\; \pi \; ɛ_{0}ɛ_{r}} \right\rbrack}}},} & (14)\end{matrix}$

where E_(f)=h-ω−E_(A) is the final (free) hole energy and c=c₀/√{squareroot over (∈_(r))} is the speed of light in silicon (∈_(r) ^(Si)≃11.9).Since E_(A)<<E_(f), the total cross section is suppressed as a ∝1/E_(f)^(3.5) (final state energy suppression). Given η_(c) photons in a cavityvolume V≃≃dλ², the acceptor lifetime is given byτ_(phot)=2V/(η_(c)cσ_(phot)) for a maximum photon-acceptor overlap. Thislimits the ability to perform active photon (sideband) cooling of thephononic cavity. However, by placing the acceptor close to a node of thephoton cavity, the ionization lifetime can increase considerably.

In embodiments, the quantum TLS can be formed in a crystalline hostformed of carbon. In particular, the host can take the form ofcrystalline carbon with the quantum TLS replacing one of the carbonatoms in the host. As used herein, the term crystalline carbon isintended to refer to those allotropes of carbon that have an organizedcrystal structure over a substantial portion of the material, forexample, diamond, lonsdaleite, a carbon nanotube, a graphene sheet, orfullerene. Referring to FIG. 16, a simplified structure 1600 ofcrystalline carbon is shown for illustrative purposes. Such structure1600 may represent a portion of a wall of a carbon nanotube, a portionof a wall of a fullerene, a portion of a graphene sheet, or a slice ofother allotropes of crystalline carbon. In the structure, carbon atoms1602 are bonded to each other via bonds 1604. A TLS 1606 may be formedby replacing one or more of the carbon atoms 1602 in the lattice 1600with an impurity (e.g., a donor or an acceptor), a defect, and/or acolor center. Phoniton use and generation in crystalline carbon can besimilar to that described above with respect to semiconductor and othermaterials.

Phonons may find use in some applications due to their slower speed thanother types of signals, such as photons. For example, many signalprocessing applications can use phonons to slow down optical signals ona chip. The phoniton and devices incorporating the phoniton(s) asdisclosed herein provides another tool to manipulate these sound wavesand has potential application in inter-conversion of information, bothclassical and quantum. In addition, phonons can be created more easilythan photons (for example, by using temperature alone). Moreover,because of the large strain energies per phonon in typicalsemiconductors and extremely small mode volume as compared to lightwavelengths, the devices incorporating phonitons can exhibit very largephonon-matter coupling (g) as compared to traditional cavity QEDsystems, even for relatively large phonon volumes. This leads to highlyquantum behavior as well as quantum behavior at higher temperatures,e.g., even at room temperature, for example when using a levitatingnanosphere (in such a configuration the laser/optical tweezer trappingsystem can be at room temperature even though the mechanical degree offreedom is cooled down effectively to very low temperatures via lasercooling approaches), which may be relevant for quantum informationtechnology such as simulation, entangling operations, etc. Large cavityphonon coupling can allow for low-energy or threshold-lessswitches/logic.

In addition, phoniton devices as disclosed herein can find applicationin telecommunications and computer systems. For example, cell phonetechnology currently employs surface acoustic wave modulators fordelaying optical signals. Phonitons provide another tool by which tomanipulate, delay, or amplify telecommunication signals. Moreover, thequantum phoniton devices provide a tool for manipulating, storing,and/or slowing phonons. Thus, the phoniton devices described herein canbe used as memory devices or in filtering applications.

Coherent phonons can be used for imaging (e.g., at smaller wavelength ascompared to lasers). The TLS transitions (and corresponding emittedphonons) of the phoniton devices described herein can operate in the THzfrequency and lower, and can be tuned over a relatively wide frequencyrange. Phonons in the THz range have nanometer-scale resolution. Becauseof this, devices incorporating phonitons may be suitable for imagingsmall objects, for example, for medical or security applications. Aphoniton device can be constructed to provide a coherent source ofphonons in a tunable regime of interest.

The phoniton is a new component for constructing and controllingmacroscopic artificial quantum systems based on sound. Besides singlephonon devices, systems composed of many coupled phonitons could exhibitnovel quantum many-body behavior. For example, “solid-sound” systems candemonstrate Mott insulator like states of phonons in coupled phonitoncavities. Cavity/qubit geometries such as these may also be relevant forquantum computing (QC) applications, for example, to mediateinteractions between distant qubits or inhibit decoherence. The systemsproposed here will benefit from the drive in silicon QC towards morepurified materials, perfect surfaces, and precise donor placement.Moreover, as described above, phoniton devices can be constructed forlow-threshold phonon lasing, phonon Bose-Einstein condensation sensors,and solid sound by incorporating multiple phonitons on a single device.

Further applications for the disclosed phoniton devices can include, butare not limited to, on-chip sensors, such as accelerometers,gradiometers, and sound detectors (for example, by employing phonitoncondensates); information processing, such as phonon logic; and quantuminformation processing, such as qubit conversion, delay, andentanglement.

Although specific configurations and materials have been describedherein, the teachings of the present application are not limitedthereto. Rather, other configurations and materials are also possibleaccording to one or more contemplated embodiments of the disclosedsubject matter.

The foregoing descriptions apply, in some cases, to examples generatedin a laboratory, but these examples can be extended to productiontechniques. For example, where quantities and techniques apply to thelaboratory examples, they should not be understood as limiting.

Features of the disclosed embodiments may be combined, rearranged,omitted, etc., within the scope of the invention to produce additionalembodiments. Furthermore, certain features may sometimes be used toadvantage without a corresponding use of other features.

It is, thus, apparent that there is provided, in accordance with thepresent disclosure, phoniton systems, methods, and devices. Manyalternatives, modifications, and variations are enabled by the presentdisclosure. While specific embodiments have been shown and described indetail to illustrate the application of the principles of the invention,it will be understood that the invention may be embodied otherwisewithout departing from such principles. Accordingly, Applicants intendto embrace all such alternatives, modifications, equivalents, andvariations that are within the spirit and scope of the presentinvention.

1-25. (canceled)
 26. A method for producing a phoniton, the methodcomprising: trapping a phonon in a cavity; and coupling the trappedphonon to a quantum two-level system in the cavity, the quantumtwo-level system having a first energy state and a second energy state,the phonon being coupled such that a mode of the trapped phononcorresponds to a transition at or near a difference between the firstand second energy states, the quantum two-level system being arranged atan anti-node or a node of the trapped phonon.
 27. The method of claim26, wherein the quantum two-level system is a solid-state quantumtwo-level system.
 28. The method of claim 26, wherein the quantumtwo-level system comprises at least one of a donor, an acceptor, a colorcenter, a defect, a quantum dot, a nanocrystal, or a colloidal quantumdot.
 29. The method of claim 26, wherein the coupling is such that acoupling rate between the trapped phonon mode and the two-level systemis in the strong coupling regime.
 30. The method of claim 29, whereinthe trapped phonon is coupled to the quantum two-level system such thatthe coupling energy between the phonon and the quantum two-level systemis greater than losses of the two-level system and the phonon in thecavity
 31. The method of claim 26, further comprising providing acrystalline-based device containing the quantum two-level system and theresonant cavity.
 32. The method of claim 26, wherein, prior to thetrapping, the phonon is generated external to the resonant cavity andprovided thereto.
 33. The method of claim 26, wherein, prior to thetrapping, the phonon is generated by the two-level system using at leastone of electrical generation, thermal generation, optical generation, ortunneling.
 34. The method of claim 26, further comprising altering atleast one of the first and second energy states such that a transitionamplitude or transition probability between them is changed.
 35. Themethod of claim 34, wherein the altering includes at least one ofstraining the material, applying an electric field, or applying amagnetic field.
 36. The method of claim 26, further comprising alteringa resonant frequency of the cavity such that the phonon energy in thecavity does not correspond to a transition frequency at the differencebetween the first and second energy states.
 37. The method of claim 26,wherein the phonon has a frequency in the terahertz regime or lower. 38.A method for producing a phoniton, the method comprising: (a) providinga two-level system in a phonon confinement structure, the two-levelsystem having first and second energy states, a transition between thefirst and second energy states corresponding to a particular phononmode; (b) interacting a first phonon with the two-level system so as tocause a transition of the two-level system from the first energy stateto the second energy state; (c) generating a second phonon by allowingthe two-level system to transition from the second energy state to thefirst energy state; and (d) re-directing the second phonon by way of thephonon confinement structure so as to interact with the two-level systemto cause another transition of the two-level system from the firstenergy state to the second energy state.
 39. The method of claim 38,wherein the two-level system comprises at least one of an impurity, adonor, an acceptor, a color center, a defect, and a quantum dot.
 40. Themethod of claim 38, further comprising repeating (c) and (d).
 41. Themethod of claim 38, wherein each of the first and second phonons has afrequency in the terahertz regime or lower.
 42. The method of claim 38,wherein each of the first and second phonons has said particular phononmode in the phonon confinement structure.
 43. The method of claim 38,wherein the phonon confinement structure comprises a reflector adjacentto a cavity containing the two-level system, and the re-directingincludes reflecting the second phonon back to the two-level system byway of a reflector adjacent to the cavity.
 44. The method of claim 38,wherein the phonon confinement structure comprises an array of holes ina crystalline host material containing the two-level system, and there-directing includes reflecting the second phonon back to the two-levelsystem by way of the array of holes.
 45. The method of claim 38, whereinthe phonon confinement structure comprises a phonon cavity or mechanicalcavity in a solid-state host material.